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First factor out the greatest common factor from each term. Then factor the remaining polynomial. Refer to Examples 9 and $10 .$$$6 x^{2}+24 x+18$$...

Question

First factor out the greatest common factor from each term. Then factor the remaining polynomial. Refer to Examples 9 and $10 .$$$6 x^{2}+24 x+18$$

First factor out the greatest common factor from each term. Then factor the remaining polynomial. Refer to Examples 9 and $10 .$ $$ 6 x^{2}+24 x+18 $$



Answers

Factor each polynomial. $$ x^{2}+6 x+8 $$

So when I'm doing this when the first thing I look at is that I can split this eight at right here into six, plus the positive six X plus two acts and I can use grouping for this one. So I want to Really? That one. That way, After six x squared, classic sex cost to absolute, there was expanded acts in this XX postulates quite to. And now I'm going to group the 1st 2 terms together and group the last two terms together. Men factor out in each of them. So when I look at the 1st 2 terms, what they have in common, I can pull the sticks out and I can pull a nextel. And then I'm left with ec, and I'm left with plus one right. And I look at the green one up here. What can I pull out of the second of the last two parts? Okay. I can pull a two hour. I can't pull a variable out the place to does not have one, but I can pull out the push to there. Okay, so if I pull out the positive two, I'm left with X push. So from here what I can do. Whatever's inside of the parentheses is one part of our ancestor stuff, X plus one. And then what I can do is I can take the next to what? You're six X plus two. And that's the next part of my answer here. I can also pull out the to hear B. I wanted in the most factor herbal form. I can keep this simplified here, and I can pull a two out of there. You gotta pull it to out. Remember, doesn't when you're multiplying, the order does not matter. So I can pull that to to the very front. I still have X plus one here. Now, after I pulled the two out of this one, I have three X costs one and then from here, if I boiled out these two, um, expressions right here that they're X plus one on the X plus one. And then I took the polynomial that I get and multiply that by two bad poured out. Then I will come out with the polynomial I started with. So we know that this is there an answer here. So we have two times x plus one times three play three x placed one right here.

All right. So, defector, this problem we're gonna be using factoring by grouping. So in order to do that, we're first going to need to break up our middle term to know what we want to break it up into. We're going to want to find two numbers that first multiply to the product product of our first and third term. So they need to multiply it to negative 16. They need to add to our second term, which is our six. So two numbers that would satisfy this are, um, positive. Eight and negative too, is negative. Eight times negative. Two gives this negative 16 8 plus negative, too obsessed Six. So we can rewrite the original expression as a X squared. Both are first term X minus two acts minus two. Now, we can continue with the factoring by grouping. You can see in the first, uh, the left hand side of the expression. Both of our elements haven't made in the next. We're gonna go ahead and pull out it x, don't leave behind a single axe in just a one. And then on the right hand side, you can see both of the elements have in negative to that we can take out. So it'll be in minus two. This will be finding X and then just a one. So now you can see that both of our expression the left hand side in the right hand side has X plus one in parentheses. So both attacks and negative two or being multiplied by the same thing. This allows us to rewrite our expression as eight X minus two times the expression that they're both multiplied by the X plus one. And this is our solution. We've now factored the original expression.

Given polynomial is nine x to the poor four plus 18 x cubed plus six x script. Hugh three is, though greatest in teacher That device. 9 18 on six on the variable raised to the smallest exponents is extra square, so three X square is the G C F off. Given Paul Normal nine x to the power four place 18 X cube less six X square. Now we will express each term as the product off G C F on its other factor so we can write it as three extra square multiplied by three X square. We will get nine x to the power for or when we divide nine x to the power four by three X square We will get three x square bliss three x square multiplied by six x Loose three X squared multiplied by two we will get six X square here three X squared is though common factor in all three terms. Now factor out GF that is three x square So three x square multiplied by three x square Place six X place so

Okay, so given X squared minus six X plus nine. We're gonna factor this. Remember, there's a one here so would be one times knowing is nine. And then I got a negative 60 capital A bomb. Someone over to numbers that multiple. You give me nine and combined to give me negative six. So I think negative three times negative three is nine and negative. Three combined with negative three is six. Given this and that, this is a one we can go ahead and write thesis factors. So the X minus three x minus three. Since those are the same, I can go ahead and write this as X minus three square.


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